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C*algebra
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{{Use American Englishdate=January 2019}}{{Short descriptionTopological complex vector space}}{{refimprovedate=February 2013}}Câˆ—algebras (pronounced "Cstar") are subjects of research in functional analysis, a branch of mathematics. A C*algebra is a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:  the content below is remote from Wikipedia
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 A is a topologically closed set in the norm topology of operators.
 A is closed under the operation of taking adjoints of operators.
Abstract characterization
We begin with the abstract characterization of C*algebras given in the 1943 paper by Gelfand and Naimark.A C*algebra, A, is a Banach algebra over the field of complex numbers, together with a map x mapsto x^* for xin A with the following properties: It is an involution, for every x in A:
x^{**} = (x^*)^* = x
 For all x, y in A:
(x + y)^* = x^* + y^*
(x y)^* = y^* x^*
 For every complex number Î» in C and every x in A:
(lambda x)^* = overline{lambda} x^* .
 For all x in A:
x^* x  = xx^*.
=  ^2,which is sometimes called the B*identity. For history behind the names C* and B*algebras, see the (#Some_history:_B.2Aalgebras_and_C.2Aalgebrashistory) section below.The C*identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C*norm is uniquely determined by the algebraic structure:
x^2 = x^* x = sup{lambda : x^* x  lambda ,1 text{ is not invertible} }.
pi(x y) = pi(x) pi(y) ,
pi(x^*) = pi(x)^* ,
Some history: B*algebras and C*algebrasThe term B*algebra was introduced by C. E. Rickart in 1946 to describe Banach *algebras that satisfy the condition:
Structure of C*algebrasC*algebras have a large number of properties that are technically convenient. Some of these properties can be established by using the continuous functional calculus or by reduction to commutative C*algebras. In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism.Selfadjoint elementsSelfadjoint elements are those of the form x=x*. The set of elements of a C*algebra A of the form x*x forms a closed convex cone. This cone is identical to the elements of the form xx*. Elements of this cone are called nonnegative (or sometimes positive, even though this terminology conflicts with its use for elements of R.)The set of selfadjoint elements of a C*algebra A naturally has the structure of a partially ordered vector space; the ordering is usually denoted â‰¥. In this ordering, a selfadjoint element x of A satisfies x â‰¥ 0 if and only if the spectrum of x is nonnegative,{{clarifyUnder the Gelfand representation?date=August 2014}} if and only if x = s*s for some s. Two selfadjoint elements x and y of A satisfy x â‰¥ y if xâˆ’y â‰¥ 0.This partially ordered subspace allows the definition of a positive linear functional on a C*algebra, which in turn is used to define the states of a C*algebra, which in turn can be used to construct the spectrum of a C*algebra using the GNS construction.Quotients and approximate identitiesAny C*algebra A has an approximate identity. In fact, there is a directed family {eÎ»}Î»âˆˆI of selfadjoint elements of A such that
x e_lambda rightarrow x
0 leq e_lambda leq e_mu leq 1quad mbox{ whenever } lambda leq mu.
In case A is separable, A has a sequential approximate identity. More generally, A will have a sequential approximate identity if and only if A contains a strictly positive element, i.e. a positive element h such that hAh is dense in A.
Using approximate identities, one can show that the algebraic quotient of a C*algebra by a closed proper twosided ideal, with the natural norm, is a C*algebra.Similarly, a closed twosided ideal of a C*algebra is itself a C*algebra.ExamplesFinitedimensional C*algebrasThe algebra M(n, C) of n Ã— n matrices over C becomes a C*algebra if we consider matrices as operators on the Euclidean space, Cn, and use the operator norm · on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In fact, all C*algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The selfadjoint requirement means finitedimensional C*algebras are semisimple, from which fact one can deduce the following theorem of Artinâ€“Wedderburn type:Theorem. A finitedimensional C*algebra, A, is canonically isomorphic to a finite direct sum
A = bigoplus_{e in min A } A e
where min A is the set of minimal nonzero selfadjoint central projections of A.Each C*algebra, Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(e), C). The finite family indexed on min A given by {dim(e)}e is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finitedimensional C*algebra. In the language of Ktheory, this vector is the positive cone of the K0 group of A.A â€ algebra (or, more explicitly, a â€ closed algebra) is the name occasionally used in physicsJohn A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." Quantum Information Processing. Volume 2, Number 5, pp. 381–419. Oct 2003. for a finitedimensional C*algebra. The dagger, â€ , is used in the name because physicists typically use the symbol to denote a Hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) â€ algebras feature prominently in quantum mechanics, and especially quantum information science.An immediate generalization of finite dimensional C*algebras are the approximately finite dimensional C*algebras.C*algebras of operatorsThe prototypical example of a C*algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H â†’ H. In fact, every C*algebra, A, is *isomorphic to a normclosed adjoint closed subalgebra of B(H) for a suitable Hilbert space, H; this is the content of the Gelfandâ€“Naimark theorem.C*algebras of compact operatorsLet H be a separable infinitedimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). It is also closed under involution; hence it is a C*algebra.Concrete C*algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*algebras:Theorem. If A is a C*subalgebra of K(H), then there exists Hilbert spaces {Hi}iâˆˆI such that
A cong bigoplus_{i in I } K(H_i),
where the (C*)direct sum consists of elements (Ti) of the Cartesian product Î K(Hi) with Ti â†’ 0.Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be developed. To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that H = l2. For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices k â‰¤ n and let en be the orthogonal projection onto Hn. The sequence {en}n is an approximate identity for K(H).K(H) is a twosided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H) by K(H) is the Calkin algebra.Commutative C*algebrasLet X be a locally compact Hausdorff space. The space C_0(X) of complexvalued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*algebra C_0(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C_0(X) has a multiplicative unit element if and only if X is compact. As does any C*algebra, C_0(X) has an approximate identity. In the case of C_0(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_K be a function of compact support which is identically 1 on K. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. Any such sequence of functions {f_K} is an approximate identity.The Gelfand representation states that every commutative C*algebra is *isomorphic to the algebra C_0(X), where X is the space of characters equipped with the weak* topology. Furthermore, if C_0(X) is isomorphic to C_0(Y) as C*algebras, it follows that X and Y are homeomorphic. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.C*enveloping algebraGiven a Banach *algebra A with an approximate identity, there is a unique (up to C*isomorphism) C*algebra E(A) and *morphism Ï€ from A into E(A) which is universal, that is, every other continuous *morphism {{nowrapÏ€â€‰'â€‰: A â†’ B}} factors uniquely through Ï€. The algebra E(A) is called the C*enveloping algebra of the Banach *algebra A.Of particular importance is the C*algebra of a locally compact group G. This is defined as the enveloping C*algebra of the group algebra of G. The C*algebra of G provides context for general harmonic analysis of G in the case G is nonabelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*algebra. See spectrum of a C*algebra.Von Neumann algebrasVon Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*algebra. They are required to be closed in the weak operator topology, which is weaker than the norm topology.The Shermanâ€“Takeda theorem implies that any C*algebra has a universal enveloping W*algebra, such that any homomorphism to a W*algebra factors through it.Type for C*algebrasA C*algebra A is of type I if and only if for all nondegenerate representations Ï€ of A the von Neumann algebra Ï€(A)â€²â€² (that is, the bicommutant of Ï€(A)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations Ï€ for which Ï€(A)â€²â€² is a factor.A locally compact group is said to be of type I if and only if its group C*algebra is type I.However, if a C*algebra has nontype I representations, then by results of James Glimm it also has representations of type II and type III. Thus for C*algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.C*algebras and quantum field theoryIn quantum mechanics, one typically describes a physical system with a C*algebra A with unit element; the selfadjoint elements of A (elements x with x* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a Clinear map Ï† : A â†’ C with Ï†(u*u) â‰¥ 0 for all u âˆˆ A) such that Ï†(1) = 1. The expected value of the observable x, if the system is in state Ï†, is then Ï†(x).This C*algebra approach is used in the HaagKastler axiomatization of local quantum field theory, where every open set of Minkowski spacetime is associated with a C*algebra.See also
Notes{{reflist}}References

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